Theorems · Theorem · real analysis
isMaxOn_univ_of_deriv
∀ {f : ℝ → ℝ} {b : ℝ},
ContinuousAt f b →
DifferentiableOn ℝ f (Set.Iio b) →
DifferentiableOn ℝ f (Set.Ioi b) →
(∀ x ∈ Set.Iio b, 0 ≤ deriv f x) → (∀ x ∈ Set.Ioi b, deriv f x ≤ 0) → IsMaxOn f Set.univ bSuppose f : ℝ → ℝ is continuous at b, the derivative f' is nonnegative on Iio b and
nonpositive on Ioi b. Then f attains its maximum on ℝ at b.
- Defined in
- Mathlib.Analysis.Calculus.DerivativeTest
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 192 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites16
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Realstatement and proof · cited by 25,697
- Set.univstatement · cited by 3,945
- Set.Ioistatement and proof · cited by 1,463
- Set.Iiostatement and proof · cited by 1,166
- ContinuousAtstatement and proof · cited by 697
- derivstatement and proof · cited by 676
- DifferentiableOnstatement and proof · cited by 419
- IsMaxOnstatement · cited by 114
- convex_Iciproof · cited by 29
- antitoneOn_of_deriv_nonposproof · cited by 25
- monotoneOn_of_deriv_nonnegproof · cited by 25
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